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Isotoping Heegaard surfaces in neat positions with respect to critical distance Heegaard surfaces
Topology and its Applications 159 (2012) 1106–1114
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Topology and its Applications www.elsevier.com/locate/topol
Isotoping Heegaard surfaces in neat positions with respect to critical distance Heegaard surfaces Ayako Ido Department of Mathematics, Nara Women’s University, Kitauoyanishi-machi, Nara 630-8506, Japan
a r t i c l e
i n f o
a b s t r a c t
MSC: 57M27
Suppose a closed orientable 3-manifold M has a genus g Heegaard surface P with distance d( P ) = 2g. Let Q be another genus g Heegaard surface which is strongly irreducible. Then we show that there is a height function f : M → I induced from P such that by isotopy, Q is deformed into a position satisfying the following; (1) f | Q has 2g + 2 critical points p 0 , p 1 , . . . , p 2g +1 with f ( p 0 ) < f ( p 1 ) < · · · < f ( p 2g +1 ) where p 0 is a minimum and p 2g +1 is a maximum, and p 1 , . . . , p 2g are saddles, (2) if we take regular values r i (i = 1, . . . , 2g + 1) such that f ( p i −1 ) < r i < f ( p i ), then f −1 (r i ) ∩ Q consists of a circle if i is odd, and f −1 (r i ) ∩ Q consists of two circles if i is even. © 2011 Elsevier B.V. All rights reserved.
Keywords: Heegaard splitting Hempel distance Graphic
1. Introduction Hempel [3] introduced the concept of distance of a Heegaard splitting, and it is shown by many authors that it well represents various complexities of 3-manifolds. For example, Scharlemann and Tomova [10] show that high distance Heegaard splittings are “rigid”. More precisely: Theorem 1.1. (Corollary 4.5 of [10]) If a compact orientable 3-manifold M has a genus g Heegaard surface P with distance d( P ) > 2g, then 1. any other Heegaard surface of the same genus is isotopic to P ; 2. moreover, any Heegaard surface Q of M with 2g ( Q )d( P ) is isotopic to a stabilization or boundary stabilization of P . Recently, Berge and Scharlemann [1] have shown the following fact: Fact. If a closed orientable 3-manifold M has a genus 2 Heegaard surface P with distance d( P ) = 4, then any other genus 2 Heegaard surfaces of M is isotopic to P . Moreover there exist examples of 3-manifolds each of which admits mutually non-isotopic genus 2 Heegaard surfaces with distance 3. Hence, it seems that it is natural to call P with distance d( P ) = 2g ( P ) a critical distance Heegaard surface. Then we have the following question: Question. If a closed orientable 3-manifold M has a genus g Heegaard surface P with distance d( P ) = 2g, then are any genus g Heegaard surfaces of M isotopic to P ?
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Elsevier B.V. All rights reserved.
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In this paper, we tackle the above question. Unfortunately, we do not have the answer for the above yet. We show that any other genus g Heegaard surface Q can be isotoped to a neat position with respect to a height function induced from the genus g Heegaard surface P , if Q is not isotopic to P and d( P ) = 2g. Precisely speaking: Theorem 1.2. Suppose that a closed orientable 3-manifold M has a genus g Heegaard surface P with distance d( P ) = 2g. Let Q be another genus g Heegaard surface which is strongly irreducible. Then there is a height function f : M → I induced from P (that is, the level surfaces of f with height t (= 0, 1) are isotopic to P ) such that by isotopy, Q is deformed into a position satisfying the following: 1. f | Q has 2g + 2 critical points p 0 , p 1 , . . . , p 2g +1 with f ( p 0 ) < f ( p 1 ) < · · · < f ( p 2g +1 ) where p 0 is a minimum and p 2g +1 is a maximum, and p 1 , . . . , p 2g are saddles, 2. if we take regular values r i (i = 1, . . . , 2g + 1) such that f ( p i −1 ) < r i < f ( p i ), then f −1 (r i ) ∩ Q consists of a circle if i is odd, and f −1 (r i ) ∩ Q consists of two circles if i is even. The main tool of the proof of Theorem 1.2 is Reeb graphs derived from horizontal arcs of the Rubinstein–Scharlemann graphic (or graphic for short). Graphic is introduced by Rubinstein and Scharlemann [9] for studying Reidemeister–Singer distance of two strongly irreducible Heegaard splittings. Moreover, under certain technical conditions, Li [8] shows that there exist horizontal arcs disjoint from the union of the regions each of which is labeled X , x, Y or y (for the definitions, see Section 3) of the graphic. He gives an alternative proof of Theorem 1.1 by using such horizontal arcs. In [4], to give a more detailed treatment for such horizontal arc, we introduce Reeb graphs derived from such horizontal arcs, and by introducing a method of assigning positive integer to each edge of Reeb graph, we give an estimation of Hempel distance (see Section 4). In this paper, by using Reeb graphs and the assigning method, we prove Theorem 1.2, and we hope that this could help giving the answer to the above question. 2. Preliminaries 2.1. Heegaard splittings A genus g ( 1) handlebody H is the boundary sum of g copies of a solid torus. Note that H is homeomorphic to the closure of a regular neighborhood of some finite graph Σ in R3 . The image Σ of the graph is called a spine of H . By a technical reason, throughout this paper, we suppose that each vertex of spines of genus g (> 1) handlebodies is of valency three (for a detailed discussion, see Section 2 of [7]). Let M be a closed orientable 3-manifold. We say that A ∪ P B is a (genus g) Heegaard splitting of M if A, B are genus g handlebodies in M such that M = A ∪ B and A ∩ B = ∂ A = ∂ B = P . Then P is called a (genus g) Heegaard surface of M. A disk D properly embedded in a handlebody H is called a meridian disk of H if ∂ D is an essential simple closed curve in ∂ H . If there are meridian disks D A , D B in A , B respectively so that ∂ D A = ∂ D B , A ∪ P B is said to be reducible. If there are meridian disks D A , D B in A , B respectively so that ∂ D A , ∂ D B are disjoint on P , A ∪ P B is said to be weakly reducible. It is easy to see that if a Heegaard splitting M = A ∪ P B is reducible, it is weakly reducible. If A ∪ P B is not weakly reducible, it is said to be strongly irreducible. 2.2. Curve complexes Let S be a closed connected orientable surface S of genus at least two, and C ( S ) the 1-skeleton of Harvey’s complex of essential simple closed curves on S (see [2]), that is, C ( S ) denotes the graph whose 0-simplices are isotopy classes of essential simple closed curves and whose 1-simplices connect distinct 0-simplices with disjoint representatives. We remark that C ( S ) is connected. Let x, y be 0-simplices of C ( S ). Then we define the distance between x and y, denoted by d S (x, y ), as the minimum of such d that there is a path in C ( S ) with d 1-simplices joining x and y. Let X , Y be subsets of the 0-simplices of C ( S ). Then we define
d S ( X , Y ) = min d S (x, y ) x ∈ X , y ∈ Y . Suppose that S is the boundary of a handlebody V . Then M( V ) denotes the subset of C ( S ) consisting of the 0-simplices with representatives bounding meridian disks of V . For a genus g ( 2) Heegaard splitting A ∪ P B, its Hempel distance, denoted by d( P ), is defined to be d P (M( A ), M( B )). 3. Rubinstein–Scharlemann graphics Let M be a smooth closed orientable 3-manifold. A sweep-out is a smooth map f : M → I such that for each x ∈ (0, 1), the level set f −1 (x) is a closed surface, and f −1 (0) (resp. f −1 (1)) is a connected, finite graph such that each vertex has valency three. Each of f −1 (0) and f −1 (1) is called a spine of the sweep-out. It is easy to see that each level surface of f is a Heegaard surface of M and the spines of the sweep-outs are spines of the two handlebodies in the Heegaard splitting. Conversely, given a Heegaard splitting A ∪ P B of M, it is easy to see that there is a sweep-out f of M such that each level surface of f is isotopic to P , f −1 (0) is a spine of A, and f −1 (1) is a spine of B. We call it a sweep-out obtained from A ∪ P B.
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Given two sweep-outs, f and g of M, we consider their product f × g (that is, ( f × g )(x) = ( f (x), g (x))), which is a smooth map from M to I × I . Kobayashi and Saeki [7] have shown that by arbitrarily small deformations of f and g, we can suppose that f × g is a stable map on the complement of the four spines. At each point in the complement of the spines, the differential of the map f × g is a linear map from R3 to R2 . This map have a one-dimensional kernel for a generic point in M. The discriminant set for f × g is the set of points where the differential has a higher-dimensional kernel. Mather’s classification of stable maps [6] implies that at each point of the discriminant set, the dimension of the kernel of the differential is two, and the discriminant set is a one-dimensional smooth submanifold in the complement of the spines in M. Moreover the discriminant set consists of all such x that a level surface of f is tangent to a level surface of g at x (here, we note that the tangent point is either a “center” or “saddle”). Let f , g be as above with f × g stable. The image of the discriminant set is a graph in I × I , which is called the Rubinstein–Scharlemann graphic. It is known that the Rubinstein–Scharlemann graphic is a finite 1-complex Γ with each vertex having valency four or two. Each valency four vertex is called a crossing vertex, and each valency two vertex is called a birth-death vertex. There are valency one or two vertices of the graphic on the boundary of I × I . Each component of the complement of Γ in I × I is called a region. At each point of a region, the corresponding level surfaces of f and g are disjoint or intersect transversely. The stable map f × g is generic if each arc {s} × I or I × {t } contains at most one vertex of the graphic. By Proposition 6.14 of [7], by arbitrarily small deformation of f and g, we may suppose that f × g is generic. 3.1. Labeling regions of the graphic Let f and g be sweep-outs obtained from Heegaard splittings A ∪ P B, X ∪ Q Y , respectively with f × g stable. Now we introduce how to label each region of the graphic with following the convention of [9]. For each s ∈ I , we put that P s = f −1 (s) (in particular, Σ A denotes P 0 and Σ B denotes P 1 ), A s = f −1 ([0, s]) and B s = f −1 ([s, 1]). Similarly, for t ∈ I , we put that Q t = g −1 (t ) (in particular, Σ X denotes Q 0 and ΣY denotes Q 1 ), X t = g −1 ([0, t ]) and Y t = g −1 ([t , 1]). Let (s, t ) be a point in a region of the graphic. Then either P s ∩ Q t = φ , or P s and Q t intersect transversely in a collection C = {c 1 , . . . , cn } of simple closed curves. Definition 3.1. Let C = {c 1 , . . . , cn } be as above. Then C P denotes the subset of C consisting of the elements which are essential on P s . Furthermore the subset C A of C P is defined by:
C A = c ∈ C P c bounds a disk D in Q t \ C P such that N (∂ D , D ) ⊂ A s , where N (∂ D , D ) is a regular neighborhood of ∂ D in D. Analogously C B ⊂ C P and C X , C Y ⊂ C Q are defined. If C A (resp. C B , C X , C Y ) is non-empty, the region is labeled A (resp. B, X , Y ). If C P and C Q are both empty and A s (resp. B s ) contains an essential curve of Q t , then the region is labeled b (resp. a). If C P and C Q are both empty and X t (resp. Y t ) contains an essential curve of P s , then the region is labeled y (resp. x). R A (resp. R B , R X , R Y , R a , R b , R x , R y ) denotes the closure of the union of the regions labeled A (resp. B, X , Y , a, b, x, y). R φ denotes the closure of the union of the unlabeled regions. The next lemma is proved in the proof of Lemma 3.2 of [8] (see also [4], [5]). Lemma 3.2. Let M, P , Q be as above. Suppose P , Q are strongly irreducible. If Q is not isotopic to P , then there exists t ∈ (0, 1) such that I × {t } is disjoint from R X ∪ R x and R Y ∪ R y . For a proof of the next lemma, see Corollary 5.6 of [4]. Lemma 3.3. Let t be as in Lemma 3.2. There is a subarc [s0 , s1 ] × {t } ⊂ I × {t } such that:
• (s1 , t ) ∈ {an edge of the graphic}, and • for any s ∈ (s0 , s1 ), (s, t ) ∈ R φ , and for any small > 0, (s0 − , t ) ∈ R A and (s1 + , t ) ∈ R B . Then we apply the argument of the proof of Lemma 2.1 of [8] to obtain such Q from Q t by an ambient isotopy whose support is contained in f −1 ([s0 + , s1 − ]) that satisfies the following conditions. Conditions 3.4. Let Q ∗ = f −1 ([s0 + , s1 − ]) ∩ Q . Then, 1. at each s ∈ (s0 , s1 ), Q ∗ is transverse to each P s , except for finitely many critical levels x1 , . . . , xn ∈ (s0 , s1 ); 2. at each critical level xi , Q ∗ is transverse to P xi except for a saddle or circle tangency, as shown in Fig. 2.1(a) of [8]; 3. at each regular level s ∈ (s0 , s1 ), each component of P s ∩ Q ∗ is a simple closed curve which is essential in both P s and Q ∗ . In the remainder of this paper, we abuse notion by denoting Q for Q .
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Fig. 1. Assigning an integer to each edge of G.
4. An estimation of Hempel distance by Reeb graph We continue with Section 3. Particularly, let s0 , s1 be as in Lemma 3.3, and Q ∗ as in Conditions 3.4. Define the equivalence relation ∼ on points on Q ∗ by x ∼ y whenever x, y are in the same component of a level set of f | Q ∗ : Q ∗ → [s0 + , s1 − ]. The Reeb graph corresponding to f | Q ∗ is the quotient space of Q ∗ by the relation ∼. Then G denotes the Reeb graph corresponding to f | Q ∗ . Note that G is a finite 1-complex such that the edges of G come from annuli in Q ∗ fibered by level loops, and that the valency two vertices correspond to circle tangencies, the valency three vertices correspond to saddle tangencies, and the valency one vertices correspond to components of ∂ Q ∗ . In particular, if a valency one vertex corresponds to a component of f −1 (s0 + ) ∩ Q (resp. f −1 (s1 − ) ∩ Q ), then it is called a ∂− -vertex (resp. ∂+ -vertex). The union of ∂− -vertices (resp. ∂+ -vertices) is denoted by ∂− G (resp. ∂+ G). Let f ∗ : G → [s0 + , s1 − ] be the function induced from f | Q ∗ . Note that for each s ∈ (s0 + , s1 − ), f ∗−1 (s)(⊂ G ) consists of a finite number of points corresponding to the components of P s ∩ Q ∗ . In [4], a method of estimating Hempel distance by using Reeb graph is given. We note that the Reeb graphs in [4] are slightly different from the above Reeb graphs, since circle tangencies did not come to appear in [4]. However it is easy to see that the arguments in [4] work in the setting of this paper. We introduce it here. Let G, ∂± G, f ∗ be as above. We assign a positive integer to each edge of G according to the following steps. Let w 1 , . . . , w k be the vertices of G which are not ∂ -vertices. We suppose that w 1 , . . . , w k are positioned in this order from the left, i.e., f ∗ ( w 1 ) < f ∗ ( w 2 ) < · · · < f ∗ ( w k ). Now we define Steps 0, 1 and 2 inductively for assigning positive integers to the edges of G (see Fig. 1). Step 0. We assign 1 to every edge adjacent to ∂− G. Step 1. Suppose that there is a valency two vertex w i adjacent to edges el , el such that el has already been assigned and el has not been assigned yet. Then we assign the same integer as that of el to el . We apply this assignment as much as possible. In our assigning process, we will repeat applications of Steps 1 and 2. Before describing Step 2, we will give a general condition that the assignments have in the process. Suppose we finish Step 1 in repeated applications of Steps 1 and 2. At this stage, either every edge of G is assigned exactly one integer, or there is a unique vertex w i such that there is an unassigned edge adjacent to w i , and that each edge of G containing a point x with f ∗ (x) < f ∗ ( w i ) has already been
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assigned exactly one integer. Then we suppose that the assigned integers satisfy the following condition (∗). (Note that the conditions are clearly satisfied after Steps 0 and 1.) (∗) For a small > 0, let L i be the set of the edges of G each containing a point x with f ∗ (x) = f ∗ ( w i ) − . Then it satisfies one of the following conditions: (1) All of the elements of L i are assigned with the same integer, say n. (2) The set of the integers assigned to the elements of L i consists of consecutive two integers, say n − 1 and n. Step 2. 1. Suppose that the vertex w i satisfies the condition (1). Then we assign n + 1 to the unassigned edge(s) adjacent to w i . 2. Suppose that the vertex w i satisfies the condition (2). Then we assign n to the unassigned edge(s) adjacent to w i . After finishing Step 2, we apply Step 1. Here we note that there are no multiple assignments. If all of the edges are assigned integers, then we are done. Suppose there is an unassigned edge. Then there is a unique vertex w j such that there is an unassigned edge adjacent to w j , and that each edge of G that contains a point y such that f ∗ ( y ) < f ∗ ( w j ) has already been assigned. Then we can show that the assignments at the current stage, also satisfies (∗) (for the proof, see Lemma 7.1 of [4]). And by repeating the processes, we finally assign an integer to each edge of G. The next theorem is proved as in the proof of Theorem 7.3 of [4]. Theorem 4.1. Let P , Q and G be as above. Let n be the minimum of the integers assigned to the edges adjacent to ∂+ G. Then the distance d( P ) is at most n + 1. 5. Proof of main theorem Let P , Q , Q ∗ , G, ∂± G, f , f ∗ be as in Section 4. In this section, we suppose that P and Q are genus g Heegaard surfaces and d( P ) = 2g, and then we consider the Reeb graph G derived from Q ∗ . Since Q is connected, there is a component of Q ∗ , say Qˆ ∗ , whose Reeb graph contains a ∂− -vertex and a ∂+ -vertex. Let Gˆ ∗ be the Reeb graph corresponding to Qˆ ∗ . Claim 5.1. The Reeb graph Gˆ ∗ contains exactly 2g − 2 valency three vertices, and each component of cl( Q \ Qˆ ∗ ) is an annulus. Proof. By 3 of Condition 3.4, each component of ∂ Qˆ ∗ is essential on Q . Hence by Euler characteristic argument, we see that Qˆ ∗ contains at most 2g − 2 saddles. By the rule in the assigning process, for each valency three vertex v, the difference in the integers assigned to the edges adjacent to v is at most one. Hence the integer assigned to the edges of Gˆ ∗ containing ∂+ vertex is at most 2g − 1. This fact together with Theorem 4.1 and the assumption d( P ) = 2g show that each edge containing ∂+ -vertex is assigned 2g − 1. Moreover by the rule in the assigning process, we see that Qˆ ∗ contains exactly 2g − 2 saddles. Hence χ ( Q \ Qˆ ∗ ) = χ ( Q ) − χ ( Qˆ ∗ ) = 0, and this implies that each component of cl( Q \ Qˆ ∗ ) is an annulus. 2 Let G be any path in the Reeb graph Gˆ ∗ joining a ∂− -vertex and a ∂+ -vertex. Claim 5.2. The path G contains all of the valency three vertices in Gˆ ∗ . Proof. Assume that G does not contain all of the valency three vertices. Then by Claim 5.1, G contains at most 2g − 3 valency three vertices. This fact together with the rule in the assigning process show that the edge of G adjacent to an ∂+ -vertex is assigned an integer less than 2g − 1. Then Theorem 4.1 shows that d( P ) < 2g, a contradiction. Hence this claim holds. 2 Let v 1 , . . . , v 2g −2 be the valency three vertices contained in G . By the proof of Claim 5.2, we see that by changing subscripts if necessary, we may suppose for each i, the set of the integers assigned to the edges adjacent to v i consists of consecutive two integers {i , i + 1}. Claim 5.3. f ∗ ( v 1 ) < f ∗ ( v 2 ) < · · · < f ∗ ( v 2g −2 ). Proof. Suppose that there exists i such that f ∗ ( v i ) > f ∗ ( v i +1 ), then there is a point x in the path G , which joins v i +1 and a ∂+ -vertex such that f ∗ ( v i +1 ) < f ∗ (x) < f ∗ ( v i ). Note that by Claim 5.2, the edge containing x is assigned at least i + 2. Then there is a point x in the path G , which joins a ∂− -vertex and v i such that f ∗ (x) = f ∗ (x ). By Claim 5.2, the integer assigned to the edge containing x is at most i, contradicting condition (∗). 2
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Fig. 2. The Reeb graph Gˆ ∗ .
Fig. 3. i is odd.
Fig. 4. i is even.
Fig. 5. The possible patterns of assigned edges adjacent to v i .
By Claim 5.1 and the assumption d( P ) = 2g, we see that Qˆ ∗ is the unique component whose Reeb graph contains both a ∂− -vertex and a ∂+ -vertex (Fig. 2). Claim 5.4. For v i (i = 1, . . . , 2g − 2), if i is odd, then the number of edges assigned i adjacent to v i is exactly two. If i is even, then the number of edges assigned i adjacent to v i is exactly one. Proof. By the fact that each component of cl( Q \ Qˆ ∗ ) is an annulus (Claim 5.1) and 3 of Conditions 3.4, we see that each component of Q ∗ other than Qˆ ∗ is an annulus. This fact together with Theorem 4.1 and the assumption d( P ) = 2g show that both of the boundary components of each annulus are contained in f −1 (s0 + ) or f −1 (s1 − ). This shows that the number of edges containing ∂− G is even, and the number of ∂− -vertices contained in Gˆ ∗ is even. This together with the fact that v 1 is the only vertex which is adjacent to edges assigned 1, show that the number of edges assigned 1 in Gˆ ∗ adjacent to ∂− G is exactly two. This implies that the number of edges assigned 1 in Gˆ ∗ adjacent to v 1 is two and the number of edges assigned 2 adjacent to v 1 is one. This implies that the number of edges assigned 2 adjacent to v 2 is exactly one. By this fact and the fact that for each i, the set of the integers assigned to the edges adjacent to v i consists of consecutive two integers {i , i + 1}, we see that the remaining two edges are assigned 3. By repeating the arguments, we have this claim (Figs. 3 and 4). 2 We say that v i is in a normal position, if v i satisfies the following: For each edge e adjacent to v i , if e is assigned i (resp. i + 1), then for each x ∈ e, f ∗ (x) < f ∗ ( v i ) (resp. f ∗ (x) > f ∗ ( v i )). (i )
Claim 5.5. For each v i , there is an ambient isotopy ϕt
of M whose support is contained in f −1 ([ f ∗ ( v i ) − , f ∗ ( v i ) + ]) such that
ϕ1 ( Q ∗ ) satisfies Conditions 3.4, hence, the Reeb graph of ϕ1(i) ( Q ∗ ) is defined, where the Reeb graph contains exactly one valency three vertex between the levels f ∗ ( v ) − and f ∗ ( v ) + , which is in a normal position. (i )
i
i
Proof. Suppose that v i is not in a normal position. Then the possible patterns of assigned edges adjacent to v i are shown in Fig. 5 (1)–(6). We consider the pattern (1). The component of f −1 ([ f ∗ ( v i ) − , f ∗ ( v i ) + ]) ∩ Q ∗ containing the critical point of f | Q ∗ is a wedge of two circles c 1 , c 2 on Q ∗ . Here, c 1 corresponds to the edge assigned i + 1. Now we take a narrow annulus A such that A ∩ Q ∗ = ∂ A ∩ Q ∗ = c 1 , f |A has no critical point, and f (∂ A \ c 1 ) = f ∗ ( v i ) + /2. Then we push c 1 along A to (i ) deform Q ∗ as in Fig. 6. By this figure, we see that this ambient isotopy gives ϕt .
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Fig. 6. The pattern (1).
Fig. 7. The patterns (2) and (5).
Fig. 8. The patterns (3) and (6).
The pattern (4) can be treated in a similar argument as in the pattern (1). The patterns (2), (5) can be treated by two successive applications of the above arguments as described in Fig. 7. Details the left to the reader. The patterns (3), (6) can be treated by three successive applications of the above arguments as described in Fig. 8. Details the left to the reader. 2 Here, for simplicity, we use Q ∗ for
ϕ11 (ϕ12 (· · · (ϕ12g −2 ( Q ∗ )) · · ·)). (1)
Claim 5.6. There is an ambient isotopy ψt
(1) of M whose support is contained in f −1 ([s0 + , f ∗ ( v 2 )]) such that ψ1 ( Q ∗ ) satisfies
Conditions 3.4, hence, the Reeb graph of ψ1 ( Q ∗ ) is defined, and it satisfies the following. There is exactly one valency three vertex in the levels (s0 + , f ∗ ( v 2 )), which has the level f ∗ ( v 1 ) and is in a normal position, and every edge which is assigned 1 is contained in between the levels s0 + and f ∗ ( v 1 ). (1)
Proof. Let P1 , P2 , be the path in Gˆ ∗ each of which joins a ∂− -vertex and v 1 . We consider the subset Qi of Pi consisting of the point x with f ∗ (x) f ∗ ( v 1 ). Since v 1 is in a normal position (Claim 5.5), Qi is a union of subarcs contained in Int(Pi ). By condition (∗) in Section 4, we see that Qi is contained between the levels f ∗ ( v 1 ) and f ∗ ( v 2 ). Note that the preimage of each arc is an annulus properly embedded in the product region f −1 ([ f ∗ ( v 1 ), f ∗ ( v 2 )]), whose boundary is contained in the boundary component of f −1 ( f ∗ ( v 1 )). This shows that there is an ambient isotopy of M whose support is contained in a small neighborhood of f −1 ([ f ∗ ( v 1 ), f ∗ ( v 2 )]), which pushes the annuli corresponding to Q1 ∪ Q2 out of f −1 ([ f ∗ ( v 1 ), f ∗ ( v 2 )]). Note that this ambient isotopy does not affect the vertex v 1 . Hence we obtained a desired ambient isotopy (Fig. 9). 2
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(1)
Fig. 9. Ambient isotopy ψt
.
(2)
Fig. 10. Ambient isotopy ψt
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.
Here, for simplicity, we use Q ∗ for ψ1 ( Q ∗ ). By using similar arguments as in the proof of Claim 5.6, we have the following. (1)
(2)
Claim 5.7. There is an ambient isotopy ψt
of M whose support is contained in f −1 ([ f ∗ ( v 1 ), f ∗ ( v 3 )]) which satisfies Conditions 3.4,
hence, the Reeb graph of ψ1 ( Q ∗ ) is defined, and it satisfies the following. There is exactly one valency three vertex in the levels ( f ∗ ( v ), f ∗ ( v )), which has the level f ∗ ( v ) and is in a normal position, and every edge which is assigned 2 is contained in between (2)
1
2
the levels f ∗ ( v 1 ) and f ∗ ( v 2 ).
2
Then we successively apply similar argument to v 3 , v 4 , . . . , v 2g −2 , and we finally obtain the next claim (see Fig. 10). Claim 5.8. We may suppose that Q ∗ satisfies the following: 1. each v i in G ∗ is in a normal position; 2. for each i (1 i 2g − 3), each edge assigned i + 1 is contained in between the levels f ∗ ( v i ) and f ∗ ( v i +1 ). The edge assigned 1 is contained in the levels s0 + and f ∗ ( v 1 ), and the edge assigned 2g − 1 is contained in the levels f ∗ ( v 2g −2 ) and s1 − . Claim 5.9. There is an ambient isotopy Φt of M such that Φ1 ( Q ∗ ) satisfies Condition 3.4, hence, the Reeb graph of Φ1 ( Q ∗ ) is defined, and it satisfies the following. For each i (1 i 2g − 3), each edge assigned i + 1 joins v i and v i +1 , each edge assigned 1 joins a ∂− -vertex and v 1 , and each edge assigned 2g − 1 joins v 2g −2 and ∂+ -vertex. Proof. By Claim 5.8, we see that for each i (1 i 2g − 3), each component of the subset of Q ∗ between the levels f ∗ ( v i ) and f ∗ ( v i +1 ), or between the levels s0 + and f ∗ ( v 1 ), or between the levels f ∗ ( v 1 ) and s1 + , is an annulus (with pinches introduces by saddles) properly embedded in the corresponding product regions f −1 ([ f ∗ ( v i ), f ∗ ( v i +1 )]), or f −1 ([s0 + , f ∗ ( v 1 )]), or f −1 ([ f ∗ ( v 2g −2 ), s1 − ]). Hence by using ambient isotopy, we can straighten each annulus with the boundary components fixed so that no critical point exists in the interior of the annulus. The composition of the above ambient isotopes gives Φt (Fig. 11). 2 Here, for simplicity, we use Q ∗ for Φ1 ( Q ∗ ).
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Fig. 11. The Reeb graph of Qˆ ∗ .
Completion of the proof. We have obtained Q with Q ∗ satisfies the conditions in Claim 5.9. Recall that Qˆ ∗ is the component of Q ∗ joining a ∂− -vertex and a ∂+ -vertex. By Claim 5.1 and 3 of Conditions 3.4, we see that each component of Q ∗ other than Qˆ ∗ is an annulus properly embedded in the product region f −1 ([s0 + , s1 − ]). Then by the proof of Claim 5.6, we can push such annulus out of the product region. Hence we may suppose that Qˆ ∗ = Q ∗ , and this implies that cl( Q \ Qˆ ∗ ) consists of two annuli, say A− , A+ such that A− is properly embedded in the handlebody f −1 ([0, s0 + ]), and A+ is properly embedded in the handlebody f −1 ([s1 − , 1]). Assume that A− is compressible in the handlebody f −1 ([0, s0 + ]). Then by compressing A− , we see that each component of ∂ A− bounds a disk in the handlebody, and hence has distance 0 in the handlebody. This together with the saddles in Q ∗ show that d( P ) is at most 2g − 1, a contradiction. Hence A− is incompressible in the handlebody. Then we have the following two cases. Case 1. A− is not essential in the handlebody (that is, A− is ∂ -parallel in the handlebody). In this case, we can isotope A− with fixing ∂ A− so that A− is contained in a small collar neighborhood of the boundary of the handlebody, which is the union of level surfaces. Then it is easy to see that we can further isotope A− in the collar neighborhood so that the critical points of f |A− consists of one minimal point, and one saddle point. Case 2. A− is essential in the handlebody. In this case, we can ∂ -compress A− to obtain a meridian disk, say D − , in the handlebody. Then we can take a spine Σ− of the handlebody such that Σ− intersects D − transversely in a single point. Then we retake f with respect to Σ− so that Σ− ∩ D − is the unique critical point (which is obviously minimal) of f | D − . Note that A− can be recovered from
D − by adding a band. We may suppose that the band is contained in a collar neighborhood as in Case 1. Hence we may suppose that this band contains exactly one critical point which is a saddle. Hence the critical points of f |A− consists of one minimal point and one saddle point. In either case we have shown that the critical points of f |A− consists of one minimal point and one saddle point. The annulus A+ properly embedded in the handlebody f −1 ([s1 − , 1]) is treated in same manner. These fact together with Claim 5.9 completes the proof of Theorem 1.2. Acknowledgements The author would like to thank Professor Tsuyoshi Kobayashi for many helpful advices and comments. She also thanks the referee for careful reading of the first version of the paper. References [1] J. Berge, M. Scharlemann, Multiple genus 2 Heegaard splittings: A missed case, preprint, arXiv:0910.3921. [2] W.J. Harvey, Boundary structure of the modular group, Riemann surfaces and related topics, in: Proceedings of the 1978 Stony Brook Conference, State Univ. New York, Stony Brook, NY, 1978, Princeton Univ. Press, Princeton, NJ, 1981, pp. 245–251. [3] J. Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (3) (2001) 631–657. [4] A. Ido, An estimation of Hempel distance by using Reeb graph, J. Knot Theory Ramifications 20 (8) (2011) 1073–1092. [5] J. Johnson, Flipping and stabilizing Heegaard splittings, preprint, arXiv:0805.4422 [math.GT], 2008. [6] J.N. Mather, Stability of C ∞ mappings, V: Transversality, Adv. Math. 4 (1970) 301–336. [7] T. Kobayashi, O. Saeki, The Rubinstein–Scharlemann graphic of a 3-manifold as the discriminant set of a stable map, Pacific J. Math. 195 (1) (2000) 101–156. [8] T. Li, Saddle tangencies and the distance of Heegaard splittings, Algebr. Geom. Topol. 7 (2007) 1119–1134. [9] H. Rubinstein, M. Scharlemann, Comparing Heegaard splittings of non-Haken 3-manifolds, Topology 35 (4) (1996) 1005–1026. [10] M. Scharlemann, M. Tomova, Alternate Heegaard genus bounds distance, Geom. Topol. 10 (2006) 593–617.
Contents lists available at SciVerse ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Isotoping Heegaard surfaces in neat positions with respect to critical distance Heegaard surfaces Ayako Ido Department of Mathematics, Nara Women’s University, Kitauoyanishi-machi, Nara 630-8506, Japan
a r t i c l e
i n f o
a b s t r a c t
MSC: 57M27
Suppose a closed orientable 3-manifold M has a genus g Heegaard surface P with distance d( P ) = 2g. Let Q be another genus g Heegaard surface which is strongly irreducible. Then we show that there is a height function f : M → I induced from P such that by isotopy, Q is deformed into a position satisfying the following; (1) f | Q has 2g + 2 critical points p 0 , p 1 , . . . , p 2g +1 with f ( p 0 ) < f ( p 1 ) < · · · < f ( p 2g +1 ) where p 0 is a minimum and p 2g +1 is a maximum, and p 1 , . . . , p 2g are saddles, (2) if we take regular values r i (i = 1, . . . , 2g + 1) such that f ( p i −1 ) < r i < f ( p i ), then f −1 (r i ) ∩ Q consists of a circle if i is odd, and f −1 (r i ) ∩ Q consists of two circles if i is even. © 2011 Elsevier B.V. All rights reserved.
Keywords: Heegaard splitting Hempel distance Graphic
1. Introduction Hempel [3] introduced the concept of distance of a Heegaard splitting, and it is shown by many authors that it well represents various complexities of 3-manifolds. For example, Scharlemann and Tomova [10] show that high distance Heegaard splittings are “rigid”. More precisely: Theorem 1.1. (Corollary 4.5 of [10]) If a compact orientable 3-manifold M has a genus g Heegaard surface P with distance d( P ) > 2g, then 1. any other Heegaard surface of the same genus is isotopic to P ; 2. moreover, any Heegaard surface Q of M with 2g ( Q )d( P ) is isotopic to a stabilization or boundary stabilization of P . Recently, Berge and Scharlemann [1] have shown the following fact: Fact. If a closed orientable 3-manifold M has a genus 2 Heegaard surface P with distance d( P ) = 4, then any other genus 2 Heegaard surfaces of M is isotopic to P . Moreover there exist examples of 3-manifolds each of which admits mutually non-isotopic genus 2 Heegaard surfaces with distance 3. Hence, it seems that it is natural to call P with distance d( P ) = 2g ( P ) a critical distance Heegaard surface. Then we have the following question: Question. If a closed orientable 3-manifold M has a genus g Heegaard surface P with distance d( P ) = 2g, then are any genus g Heegaard surfaces of M isotopic to P ?
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In this paper, we tackle the above question. Unfortunately, we do not have the answer for the above yet. We show that any other genus g Heegaard surface Q can be isotoped to a neat position with respect to a height function induced from the genus g Heegaard surface P , if Q is not isotopic to P and d( P ) = 2g. Precisely speaking: Theorem 1.2. Suppose that a closed orientable 3-manifold M has a genus g Heegaard surface P with distance d( P ) = 2g. Let Q be another genus g Heegaard surface which is strongly irreducible. Then there is a height function f : M → I induced from P (that is, the level surfaces of f with height t (= 0, 1) are isotopic to P ) such that by isotopy, Q is deformed into a position satisfying the following: 1. f | Q has 2g + 2 critical points p 0 , p 1 , . . . , p 2g +1 with f ( p 0 ) < f ( p 1 ) < · · · < f ( p 2g +1 ) where p 0 is a minimum and p 2g +1 is a maximum, and p 1 , . . . , p 2g are saddles, 2. if we take regular values r i (i = 1, . . . , 2g + 1) such that f ( p i −1 ) < r i < f ( p i ), then f −1 (r i ) ∩ Q consists of a circle if i is odd, and f −1 (r i ) ∩ Q consists of two circles if i is even. The main tool of the proof of Theorem 1.2 is Reeb graphs derived from horizontal arcs of the Rubinstein–Scharlemann graphic (or graphic for short). Graphic is introduced by Rubinstein and Scharlemann [9] for studying Reidemeister–Singer distance of two strongly irreducible Heegaard splittings. Moreover, under certain technical conditions, Li [8] shows that there exist horizontal arcs disjoint from the union of the regions each of which is labeled X , x, Y or y (for the definitions, see Section 3) of the graphic. He gives an alternative proof of Theorem 1.1 by using such horizontal arcs. In [4], to give a more detailed treatment for such horizontal arc, we introduce Reeb graphs derived from such horizontal arcs, and by introducing a method of assigning positive integer to each edge of Reeb graph, we give an estimation of Hempel distance (see Section 4). In this paper, by using Reeb graphs and the assigning method, we prove Theorem 1.2, and we hope that this could help giving the answer to the above question. 2. Preliminaries 2.1. Heegaard splittings A genus g ( 1) handlebody H is the boundary sum of g copies of a solid torus. Note that H is homeomorphic to the closure of a regular neighborhood of some finite graph Σ in R3 . The image Σ of the graph is called a spine of H . By a technical reason, throughout this paper, we suppose that each vertex of spines of genus g (> 1) handlebodies is of valency three (for a detailed discussion, see Section 2 of [7]). Let M be a closed orientable 3-manifold. We say that A ∪ P B is a (genus g) Heegaard splitting of M if A, B are genus g handlebodies in M such that M = A ∪ B and A ∩ B = ∂ A = ∂ B = P . Then P is called a (genus g) Heegaard surface of M. A disk D properly embedded in a handlebody H is called a meridian disk of H if ∂ D is an essential simple closed curve in ∂ H . If there are meridian disks D A , D B in A , B respectively so that ∂ D A = ∂ D B , A ∪ P B is said to be reducible. If there are meridian disks D A , D B in A , B respectively so that ∂ D A , ∂ D B are disjoint on P , A ∪ P B is said to be weakly reducible. It is easy to see that if a Heegaard splitting M = A ∪ P B is reducible, it is weakly reducible. If A ∪ P B is not weakly reducible, it is said to be strongly irreducible. 2.2. Curve complexes Let S be a closed connected orientable surface S of genus at least two, and C ( S ) the 1-skeleton of Harvey’s complex of essential simple closed curves on S (see [2]), that is, C ( S ) denotes the graph whose 0-simplices are isotopy classes of essential simple closed curves and whose 1-simplices connect distinct 0-simplices with disjoint representatives. We remark that C ( S ) is connected. Let x, y be 0-simplices of C ( S ). Then we define the distance between x and y, denoted by d S (x, y ), as the minimum of such d that there is a path in C ( S ) with d 1-simplices joining x and y. Let X , Y be subsets of the 0-simplices of C ( S ). Then we define
d S ( X , Y ) = min d S (x, y ) x ∈ X , y ∈ Y . Suppose that S is the boundary of a handlebody V . Then M( V ) denotes the subset of C ( S ) consisting of the 0-simplices with representatives bounding meridian disks of V . For a genus g ( 2) Heegaard splitting A ∪ P B, its Hempel distance, denoted by d( P ), is defined to be d P (M( A ), M( B )). 3. Rubinstein–Scharlemann graphics Let M be a smooth closed orientable 3-manifold. A sweep-out is a smooth map f : M → I such that for each x ∈ (0, 1), the level set f −1 (x) is a closed surface, and f −1 (0) (resp. f −1 (1)) is a connected, finite graph such that each vertex has valency three. Each of f −1 (0) and f −1 (1) is called a spine of the sweep-out. It is easy to see that each level surface of f is a Heegaard surface of M and the spines of the sweep-outs are spines of the two handlebodies in the Heegaard splitting. Conversely, given a Heegaard splitting A ∪ P B of M, it is easy to see that there is a sweep-out f of M such that each level surface of f is isotopic to P , f −1 (0) is a spine of A, and f −1 (1) is a spine of B. We call it a sweep-out obtained from A ∪ P B.
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Given two sweep-outs, f and g of M, we consider their product f × g (that is, ( f × g )(x) = ( f (x), g (x))), which is a smooth map from M to I × I . Kobayashi and Saeki [7] have shown that by arbitrarily small deformations of f and g, we can suppose that f × g is a stable map on the complement of the four spines. At each point in the complement of the spines, the differential of the map f × g is a linear map from R3 to R2 . This map have a one-dimensional kernel for a generic point in M. The discriminant set for f × g is the set of points where the differential has a higher-dimensional kernel. Mather’s classification of stable maps [6] implies that at each point of the discriminant set, the dimension of the kernel of the differential is two, and the discriminant set is a one-dimensional smooth submanifold in the complement of the spines in M. Moreover the discriminant set consists of all such x that a level surface of f is tangent to a level surface of g at x (here, we note that the tangent point is either a “center” or “saddle”). Let f , g be as above with f × g stable. The image of the discriminant set is a graph in I × I , which is called the Rubinstein–Scharlemann graphic. It is known that the Rubinstein–Scharlemann graphic is a finite 1-complex Γ with each vertex having valency four or two. Each valency four vertex is called a crossing vertex, and each valency two vertex is called a birth-death vertex. There are valency one or two vertices of the graphic on the boundary of I × I . Each component of the complement of Γ in I × I is called a region. At each point of a region, the corresponding level surfaces of f and g are disjoint or intersect transversely. The stable map f × g is generic if each arc {s} × I or I × {t } contains at most one vertex of the graphic. By Proposition 6.14 of [7], by arbitrarily small deformation of f and g, we may suppose that f × g is generic. 3.1. Labeling regions of the graphic Let f and g be sweep-outs obtained from Heegaard splittings A ∪ P B, X ∪ Q Y , respectively with f × g stable. Now we introduce how to label each region of the graphic with following the convention of [9]. For each s ∈ I , we put that P s = f −1 (s) (in particular, Σ A denotes P 0 and Σ B denotes P 1 ), A s = f −1 ([0, s]) and B s = f −1 ([s, 1]). Similarly, for t ∈ I , we put that Q t = g −1 (t ) (in particular, Σ X denotes Q 0 and ΣY denotes Q 1 ), X t = g −1 ([0, t ]) and Y t = g −1 ([t , 1]). Let (s, t ) be a point in a region of the graphic. Then either P s ∩ Q t = φ , or P s and Q t intersect transversely in a collection C = {c 1 , . . . , cn } of simple closed curves. Definition 3.1. Let C = {c 1 , . . . , cn } be as above. Then C P denotes the subset of C consisting of the elements which are essential on P s . Furthermore the subset C A of C P is defined by:
C A = c ∈ C P c bounds a disk D in Q t \ C P such that N (∂ D , D ) ⊂ A s , where N (∂ D , D ) is a regular neighborhood of ∂ D in D. Analogously C B ⊂ C P and C X , C Y ⊂ C Q are defined. If C A (resp. C B , C X , C Y ) is non-empty, the region is labeled A (resp. B, X , Y ). If C P and C Q are both empty and A s (resp. B s ) contains an essential curve of Q t , then the region is labeled b (resp. a). If C P and C Q are both empty and X t (resp. Y t ) contains an essential curve of P s , then the region is labeled y (resp. x). R A (resp. R B , R X , R Y , R a , R b , R x , R y ) denotes the closure of the union of the regions labeled A (resp. B, X , Y , a, b, x, y). R φ denotes the closure of the union of the unlabeled regions. The next lemma is proved in the proof of Lemma 3.2 of [8] (see also [4], [5]). Lemma 3.2. Let M, P , Q be as above. Suppose P , Q are strongly irreducible. If Q is not isotopic to P , then there exists t ∈ (0, 1) such that I × {t } is disjoint from R X ∪ R x and R Y ∪ R y . For a proof of the next lemma, see Corollary 5.6 of [4]. Lemma 3.3. Let t be as in Lemma 3.2. There is a subarc [s0 , s1 ] × {t } ⊂ I × {t } such that:
• (s1 , t ) ∈ {an edge of the graphic}, and • for any s ∈ (s0 , s1 ), (s, t ) ∈ R φ , and for any small > 0, (s0 − , t ) ∈ R A and (s1 + , t ) ∈ R B . Then we apply the argument of the proof of Lemma 2.1 of [8] to obtain such Q from Q t by an ambient isotopy whose support is contained in f −1 ([s0 + , s1 − ]) that satisfies the following conditions. Conditions 3.4. Let Q ∗ = f −1 ([s0 + , s1 − ]) ∩ Q . Then, 1. at each s ∈ (s0 , s1 ), Q ∗ is transverse to each P s , except for finitely many critical levels x1 , . . . , xn ∈ (s0 , s1 ); 2. at each critical level xi , Q ∗ is transverse to P xi except for a saddle or circle tangency, as shown in Fig. 2.1(a) of [8]; 3. at each regular level s ∈ (s0 , s1 ), each component of P s ∩ Q ∗ is a simple closed curve which is essential in both P s and Q ∗ . In the remainder of this paper, we abuse notion by denoting Q for Q .
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Fig. 1. Assigning an integer to each edge of G.
4. An estimation of Hempel distance by Reeb graph We continue with Section 3. Particularly, let s0 , s1 be as in Lemma 3.3, and Q ∗ as in Conditions 3.4. Define the equivalence relation ∼ on points on Q ∗ by x ∼ y whenever x, y are in the same component of a level set of f | Q ∗ : Q ∗ → [s0 + , s1 − ]. The Reeb graph corresponding to f | Q ∗ is the quotient space of Q ∗ by the relation ∼. Then G denotes the Reeb graph corresponding to f | Q ∗ . Note that G is a finite 1-complex such that the edges of G come from annuli in Q ∗ fibered by level loops, and that the valency two vertices correspond to circle tangencies, the valency three vertices correspond to saddle tangencies, and the valency one vertices correspond to components of ∂ Q ∗ . In particular, if a valency one vertex corresponds to a component of f −1 (s0 + ) ∩ Q (resp. f −1 (s1 − ) ∩ Q ), then it is called a ∂− -vertex (resp. ∂+ -vertex). The union of ∂− -vertices (resp. ∂+ -vertices) is denoted by ∂− G (resp. ∂+ G). Let f ∗ : G → [s0 + , s1 − ] be the function induced from f | Q ∗ . Note that for each s ∈ (s0 + , s1 − ), f ∗−1 (s)(⊂ G ) consists of a finite number of points corresponding to the components of P s ∩ Q ∗ . In [4], a method of estimating Hempel distance by using Reeb graph is given. We note that the Reeb graphs in [4] are slightly different from the above Reeb graphs, since circle tangencies did not come to appear in [4]. However it is easy to see that the arguments in [4] work in the setting of this paper. We introduce it here. Let G, ∂± G, f ∗ be as above. We assign a positive integer to each edge of G according to the following steps. Let w 1 , . . . , w k be the vertices of G which are not ∂ -vertices. We suppose that w 1 , . . . , w k are positioned in this order from the left, i.e., f ∗ ( w 1 ) < f ∗ ( w 2 ) < · · · < f ∗ ( w k ). Now we define Steps 0, 1 and 2 inductively for assigning positive integers to the edges of G (see Fig. 1). Step 0. We assign 1 to every edge adjacent to ∂− G. Step 1. Suppose that there is a valency two vertex w i adjacent to edges el , el such that el has already been assigned and el has not been assigned yet. Then we assign the same integer as that of el to el . We apply this assignment as much as possible. In our assigning process, we will repeat applications of Steps 1 and 2. Before describing Step 2, we will give a general condition that the assignments have in the process. Suppose we finish Step 1 in repeated applications of Steps 1 and 2. At this stage, either every edge of G is assigned exactly one integer, or there is a unique vertex w i such that there is an unassigned edge adjacent to w i , and that each edge of G containing a point x with f ∗ (x) < f ∗ ( w i ) has already been
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assigned exactly one integer. Then we suppose that the assigned integers satisfy the following condition (∗). (Note that the conditions are clearly satisfied after Steps 0 and 1.) (∗) For a small > 0, let L i be the set of the edges of G each containing a point x with f ∗ (x) = f ∗ ( w i ) − . Then it satisfies one of the following conditions: (1) All of the elements of L i are assigned with the same integer, say n. (2) The set of the integers assigned to the elements of L i consists of consecutive two integers, say n − 1 and n. Step 2. 1. Suppose that the vertex w i satisfies the condition (1). Then we assign n + 1 to the unassigned edge(s) adjacent to w i . 2. Suppose that the vertex w i satisfies the condition (2). Then we assign n to the unassigned edge(s) adjacent to w i . After finishing Step 2, we apply Step 1. Here we note that there are no multiple assignments. If all of the edges are assigned integers, then we are done. Suppose there is an unassigned edge. Then there is a unique vertex w j such that there is an unassigned edge adjacent to w j , and that each edge of G that contains a point y such that f ∗ ( y ) < f ∗ ( w j ) has already been assigned. Then we can show that the assignments at the current stage, also satisfies (∗) (for the proof, see Lemma 7.1 of [4]). And by repeating the processes, we finally assign an integer to each edge of G. The next theorem is proved as in the proof of Theorem 7.3 of [4]. Theorem 4.1. Let P , Q and G be as above. Let n be the minimum of the integers assigned to the edges adjacent to ∂+ G. Then the distance d( P ) is at most n + 1. 5. Proof of main theorem Let P , Q , Q ∗ , G, ∂± G, f , f ∗ be as in Section 4. In this section, we suppose that P and Q are genus g Heegaard surfaces and d( P ) = 2g, and then we consider the Reeb graph G derived from Q ∗ . Since Q is connected, there is a component of Q ∗ , say Qˆ ∗ , whose Reeb graph contains a ∂− -vertex and a ∂+ -vertex. Let Gˆ ∗ be the Reeb graph corresponding to Qˆ ∗ . Claim 5.1. The Reeb graph Gˆ ∗ contains exactly 2g − 2 valency three vertices, and each component of cl( Q \ Qˆ ∗ ) is an annulus. Proof. By 3 of Condition 3.4, each component of ∂ Qˆ ∗ is essential on Q . Hence by Euler characteristic argument, we see that Qˆ ∗ contains at most 2g − 2 saddles. By the rule in the assigning process, for each valency three vertex v, the difference in the integers assigned to the edges adjacent to v is at most one. Hence the integer assigned to the edges of Gˆ ∗ containing ∂+ vertex is at most 2g − 1. This fact together with Theorem 4.1 and the assumption d( P ) = 2g show that each edge containing ∂+ -vertex is assigned 2g − 1. Moreover by the rule in the assigning process, we see that Qˆ ∗ contains exactly 2g − 2 saddles. Hence χ ( Q \ Qˆ ∗ ) = χ ( Q ) − χ ( Qˆ ∗ ) = 0, and this implies that each component of cl( Q \ Qˆ ∗ ) is an annulus. 2 Let G be any path in the Reeb graph Gˆ ∗ joining a ∂− -vertex and a ∂+ -vertex. Claim 5.2. The path G contains all of the valency three vertices in Gˆ ∗ . Proof. Assume that G does not contain all of the valency three vertices. Then by Claim 5.1, G contains at most 2g − 3 valency three vertices. This fact together with the rule in the assigning process show that the edge of G adjacent to an ∂+ -vertex is assigned an integer less than 2g − 1. Then Theorem 4.1 shows that d( P ) < 2g, a contradiction. Hence this claim holds. 2 Let v 1 , . . . , v 2g −2 be the valency three vertices contained in G . By the proof of Claim 5.2, we see that by changing subscripts if necessary, we may suppose for each i, the set of the integers assigned to the edges adjacent to v i consists of consecutive two integers {i , i + 1}. Claim 5.3. f ∗ ( v 1 ) < f ∗ ( v 2 ) < · · · < f ∗ ( v 2g −2 ). Proof. Suppose that there exists i such that f ∗ ( v i ) > f ∗ ( v i +1 ), then there is a point x in the path G , which joins v i +1 and a ∂+ -vertex such that f ∗ ( v i +1 ) < f ∗ (x) < f ∗ ( v i ). Note that by Claim 5.2, the edge containing x is assigned at least i + 2. Then there is a point x in the path G , which joins a ∂− -vertex and v i such that f ∗ (x) = f ∗ (x ). By Claim 5.2, the integer assigned to the edge containing x is at most i, contradicting condition (∗). 2
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Fig. 2. The Reeb graph Gˆ ∗ .
Fig. 3. i is odd.
Fig. 4. i is even.
Fig. 5. The possible patterns of assigned edges adjacent to v i .
By Claim 5.1 and the assumption d( P ) = 2g, we see that Qˆ ∗ is the unique component whose Reeb graph contains both a ∂− -vertex and a ∂+ -vertex (Fig. 2). Claim 5.4. For v i (i = 1, . . . , 2g − 2), if i is odd, then the number of edges assigned i adjacent to v i is exactly two. If i is even, then the number of edges assigned i adjacent to v i is exactly one. Proof. By the fact that each component of cl( Q \ Qˆ ∗ ) is an annulus (Claim 5.1) and 3 of Conditions 3.4, we see that each component of Q ∗ other than Qˆ ∗ is an annulus. This fact together with Theorem 4.1 and the assumption d( P ) = 2g show that both of the boundary components of each annulus are contained in f −1 (s0 + ) or f −1 (s1 − ). This shows that the number of edges containing ∂− G is even, and the number of ∂− -vertices contained in Gˆ ∗ is even. This together with the fact that v 1 is the only vertex which is adjacent to edges assigned 1, show that the number of edges assigned 1 in Gˆ ∗ adjacent to ∂− G is exactly two. This implies that the number of edges assigned 1 in Gˆ ∗ adjacent to v 1 is two and the number of edges assigned 2 adjacent to v 1 is one. This implies that the number of edges assigned 2 adjacent to v 2 is exactly one. By this fact and the fact that for each i, the set of the integers assigned to the edges adjacent to v i consists of consecutive two integers {i , i + 1}, we see that the remaining two edges are assigned 3. By repeating the arguments, we have this claim (Figs. 3 and 4). 2 We say that v i is in a normal position, if v i satisfies the following: For each edge e adjacent to v i , if e is assigned i (resp. i + 1), then for each x ∈ e, f ∗ (x) < f ∗ ( v i ) (resp. f ∗ (x) > f ∗ ( v i )). (i )
Claim 5.5. For each v i , there is an ambient isotopy ϕt
of M whose support is contained in f −1 ([ f ∗ ( v i ) − , f ∗ ( v i ) + ]) such that
ϕ1 ( Q ∗ ) satisfies Conditions 3.4, hence, the Reeb graph of ϕ1(i) ( Q ∗ ) is defined, where the Reeb graph contains exactly one valency three vertex between the levels f ∗ ( v ) − and f ∗ ( v ) + , which is in a normal position. (i )
i
i
Proof. Suppose that v i is not in a normal position. Then the possible patterns of assigned edges adjacent to v i are shown in Fig. 5 (1)–(6). We consider the pattern (1). The component of f −1 ([ f ∗ ( v i ) − , f ∗ ( v i ) + ]) ∩ Q ∗ containing the critical point of f | Q ∗ is a wedge of two circles c 1 , c 2 on Q ∗ . Here, c 1 corresponds to the edge assigned i + 1. Now we take a narrow annulus A such that A ∩ Q ∗ = ∂ A ∩ Q ∗ = c 1 , f |A has no critical point, and f (∂ A \ c 1 ) = f ∗ ( v i ) + /2. Then we push c 1 along A to (i ) deform Q ∗ as in Fig. 6. By this figure, we see that this ambient isotopy gives ϕt .
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Fig. 6. The pattern (1).
Fig. 7. The patterns (2) and (5).
Fig. 8. The patterns (3) and (6).
The pattern (4) can be treated in a similar argument as in the pattern (1). The patterns (2), (5) can be treated by two successive applications of the above arguments as described in Fig. 7. Details the left to the reader. The patterns (3), (6) can be treated by three successive applications of the above arguments as described in Fig. 8. Details the left to the reader. 2 Here, for simplicity, we use Q ∗ for
ϕ11 (ϕ12 (· · · (ϕ12g −2 ( Q ∗ )) · · ·)). (1)
Claim 5.6. There is an ambient isotopy ψt
(1) of M whose support is contained in f −1 ([s0 + , f ∗ ( v 2 )]) such that ψ1 ( Q ∗ ) satisfies
Conditions 3.4, hence, the Reeb graph of ψ1 ( Q ∗ ) is defined, and it satisfies the following. There is exactly one valency three vertex in the levels (s0 + , f ∗ ( v 2 )), which has the level f ∗ ( v 1 ) and is in a normal position, and every edge which is assigned 1 is contained in between the levels s0 + and f ∗ ( v 1 ). (1)
Proof. Let P1 , P2 , be the path in Gˆ ∗ each of which joins a ∂− -vertex and v 1 . We consider the subset Qi of Pi consisting of the point x with f ∗ (x) f ∗ ( v 1 ). Since v 1 is in a normal position (Claim 5.5), Qi is a union of subarcs contained in Int(Pi ). By condition (∗) in Section 4, we see that Qi is contained between the levels f ∗ ( v 1 ) and f ∗ ( v 2 ). Note that the preimage of each arc is an annulus properly embedded in the product region f −1 ([ f ∗ ( v 1 ), f ∗ ( v 2 )]), whose boundary is contained in the boundary component of f −1 ( f ∗ ( v 1 )). This shows that there is an ambient isotopy of M whose support is contained in a small neighborhood of f −1 ([ f ∗ ( v 1 ), f ∗ ( v 2 )]), which pushes the annuli corresponding to Q1 ∪ Q2 out of f −1 ([ f ∗ ( v 1 ), f ∗ ( v 2 )]). Note that this ambient isotopy does not affect the vertex v 1 . Hence we obtained a desired ambient isotopy (Fig. 9). 2
A. Ido / Topology and its Applications 159 (2012) 1106–1114
(1)
Fig. 9. Ambient isotopy ψt
.
(2)
Fig. 10. Ambient isotopy ψt
1113
.
Here, for simplicity, we use Q ∗ for ψ1 ( Q ∗ ). By using similar arguments as in the proof of Claim 5.6, we have the following. (1)
(2)
Claim 5.7. There is an ambient isotopy ψt
of M whose support is contained in f −1 ([ f ∗ ( v 1 ), f ∗ ( v 3 )]) which satisfies Conditions 3.4,
hence, the Reeb graph of ψ1 ( Q ∗ ) is defined, and it satisfies the following. There is exactly one valency three vertex in the levels ( f ∗ ( v ), f ∗ ( v )), which has the level f ∗ ( v ) and is in a normal position, and every edge which is assigned 2 is contained in between (2)
1
2
the levels f ∗ ( v 1 ) and f ∗ ( v 2 ).
2
Then we successively apply similar argument to v 3 , v 4 , . . . , v 2g −2 , and we finally obtain the next claim (see Fig. 10). Claim 5.8. We may suppose that Q ∗ satisfies the following: 1. each v i in G ∗ is in a normal position; 2. for each i (1 i 2g − 3), each edge assigned i + 1 is contained in between the levels f ∗ ( v i ) and f ∗ ( v i +1 ). The edge assigned 1 is contained in the levels s0 + and f ∗ ( v 1 ), and the edge assigned 2g − 1 is contained in the levels f ∗ ( v 2g −2 ) and s1 − . Claim 5.9. There is an ambient isotopy Φt of M such that Φ1 ( Q ∗ ) satisfies Condition 3.4, hence, the Reeb graph of Φ1 ( Q ∗ ) is defined, and it satisfies the following. For each i (1 i 2g − 3), each edge assigned i + 1 joins v i and v i +1 , each edge assigned 1 joins a ∂− -vertex and v 1 , and each edge assigned 2g − 1 joins v 2g −2 and ∂+ -vertex. Proof. By Claim 5.8, we see that for each i (1 i 2g − 3), each component of the subset of Q ∗ between the levels f ∗ ( v i ) and f ∗ ( v i +1 ), or between the levels s0 + and f ∗ ( v 1 ), or between the levels f ∗ ( v 1 ) and s1 + , is an annulus (with pinches introduces by saddles) properly embedded in the corresponding product regions f −1 ([ f ∗ ( v i ), f ∗ ( v i +1 )]), or f −1 ([s0 + , f ∗ ( v 1 )]), or f −1 ([ f ∗ ( v 2g −2 ), s1 − ]). Hence by using ambient isotopy, we can straighten each annulus with the boundary components fixed so that no critical point exists in the interior of the annulus. The composition of the above ambient isotopes gives Φt (Fig. 11). 2 Here, for simplicity, we use Q ∗ for Φ1 ( Q ∗ ).
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Fig. 11. The Reeb graph of Qˆ ∗ .
Completion of the proof. We have obtained Q with Q ∗ satisfies the conditions in Claim 5.9. Recall that Qˆ ∗ is the component of Q ∗ joining a ∂− -vertex and a ∂+ -vertex. By Claim 5.1 and 3 of Conditions 3.4, we see that each component of Q ∗ other than Qˆ ∗ is an annulus properly embedded in the product region f −1 ([s0 + , s1 − ]). Then by the proof of Claim 5.6, we can push such annulus out of the product region. Hence we may suppose that Qˆ ∗ = Q ∗ , and this implies that cl( Q \ Qˆ ∗ ) consists of two annuli, say A− , A+ such that A− is properly embedded in the handlebody f −1 ([0, s0 + ]), and A+ is properly embedded in the handlebody f −1 ([s1 − , 1]). Assume that A− is compressible in the handlebody f −1 ([0, s0 + ]). Then by compressing A− , we see that each component of ∂ A− bounds a disk in the handlebody, and hence has distance 0 in the handlebody. This together with the saddles in Q ∗ show that d( P ) is at most 2g − 1, a contradiction. Hence A− is incompressible in the handlebody. Then we have the following two cases. Case 1. A− is not essential in the handlebody (that is, A− is ∂ -parallel in the handlebody). In this case, we can isotope A− with fixing ∂ A− so that A− is contained in a small collar neighborhood of the boundary of the handlebody, which is the union of level surfaces. Then it is easy to see that we can further isotope A− in the collar neighborhood so that the critical points of f |A− consists of one minimal point, and one saddle point. Case 2. A− is essential in the handlebody. In this case, we can ∂ -compress A− to obtain a meridian disk, say D − , in the handlebody. Then we can take a spine Σ− of the handlebody such that Σ− intersects D − transversely in a single point. Then we retake f with respect to Σ− so that Σ− ∩ D − is the unique critical point (which is obviously minimal) of f | D − . Note that A− can be recovered from
D − by adding a band. We may suppose that the band is contained in a collar neighborhood as in Case 1. Hence we may suppose that this band contains exactly one critical point which is a saddle. Hence the critical points of f |A− consists of one minimal point and one saddle point. In either case we have shown that the critical points of f |A− consists of one minimal point and one saddle point. The annulus A+ properly embedded in the handlebody f −1 ([s1 − , 1]) is treated in same manner. These fact together with Claim 5.9 completes the proof of Theorem 1.2. Acknowledgements The author would like to thank Professor Tsuyoshi Kobayashi for many helpful advices and comments. She also thanks the referee for careful reading of the first version of the paper. References [1] J. Berge, M. Scharlemann, Multiple genus 2 Heegaard splittings: A missed case, preprint, arXiv:0910.3921. [2] W.J. Harvey, Boundary structure of the modular group, Riemann surfaces and related topics, in: Proceedings of the 1978 Stony Brook Conference, State Univ. New York, Stony Brook, NY, 1978, Princeton Univ. Press, Princeton, NJ, 1981, pp. 245–251. [3] J. Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (3) (2001) 631–657. [4] A. Ido, An estimation of Hempel distance by using Reeb graph, J. Knot Theory Ramifications 20 (8) (2011) 1073–1092. [5] J. Johnson, Flipping and stabilizing Heegaard splittings, preprint, arXiv:0805.4422 [math.GT], 2008. [6] J.N. Mather, Stability of C ∞ mappings, V: Transversality, Adv. Math. 4 (1970) 301–336. [7] T. Kobayashi, O. Saeki, The Rubinstein–Scharlemann graphic of a 3-manifold as the discriminant set of a stable map, Pacific J. Math. 195 (1) (2000) 101–156. [8] T. Li, Saddle tangencies and the distance of Heegaard splittings, Algebr. Geom. Topol. 7 (2007) 1119–1134. [9] H. Rubinstein, M. Scharlemann, Comparing Heegaard splittings of non-Haken 3-manifolds, Topology 35 (4) (1996) 1005–1026. [10] M. Scharlemann, M. Tomova, Alternate Heegaard genus bounds distance, Geom. Topol. 10 (2006) 593–617.